Sharedwww / lectures / day4.dviOpen in CoCalc
����;� TeX output 2004.02.17:0927�������a����L���C����<ݍ�N'�src:15 day4.tex�D��t�qG�cmr17�Math�B�129:�m�Algebraic�Num��uNb���er�Theory����������N��qcmbx12�Lecture��
4��#�􍍍�������X�Qffcmr12�William��/Stein������&a���v	T���4uesda���y�,��/F�ebruary�17,�2004��-�%��src:17 day4.tex�X�Qcmr12�Note:��There's���a�b�S�o�ok���called�����@cmti12�A��2lgebr��ffaic�SNumb�er�The�ory�and�F���ermat's�L�ast�The�or�em�������b��ry�~yStew�art�and�T��Vall,��nwhic�h�app�S�ears�to�ha�v�e�a�detailed�in�tro�S�duction�to�algebraic�����n��rum�b�S�er�
%theory�and�assumes�little�bac��rkground�on�the�part�of�the�reader.��VThere�is�a�����discussion��Kof�the�denition�of�mo�S�dule,���and�pro�ofs�of�basic�facts�ab�out�n��rum�b�er��Kelds,�����and���man��ry�exercises.�c�If�y�ou�nd�Swinnerton-Dy�er's�b�S�o�ok���dicult,���y�ou�migh�t�w�an�t�����to���try�to�get�y��rour�hands�on�Stew�art�and�T��Vall,�Ϡwhic�h�costs�ab�S�out�$38�new.�^f(Hand�����around��a�cop��ry��V.)����% ��src:26 day4.texT��Vo�S�da��ry���w�e�will�deduce,�� with�complete�pro�S�ofs,�the�most�imp�S�ortan��rt�basic�prop�ert��ry�����of��Kthe�ring�of�in��rtegers��src:27 day4.tex�"!",� cmsy10�O����� �2cmmi8�K�� � �of�an�algebraic�n�um�b�S�er,��^namely�that�ev�ery�nonzero�ideals�����can���b�S�e�written�uniquely�as�pro�ducts�of�prime�ideals.��After�pro��rving�this�fundamen�tal�����theorem,���w��re��will�compute�some�examples�using�MA�GMA.�On�Th�ursda�y�the�lecture�����will��mconsist�mostly�of�examples�illustrating�the�substan��rtial�theory�w�e�will�ha�v�e�already�����dev��relop�S�ed,��so�hang�in�there!��(V����src:34 day4.tex��/��N�G� cmbx12�1�� (�Dedekind�z�Domains��b#����0��N� cmbx12�Corollary��1.1.��src:36 day4.tex�The�35ring�of�inte��ffgers��O�����K�� n��of�a�numb�er� eld�is�No�etherian.��������Pr��ffo�of.���8I�src:39 day4.tex�As��Yw��re�sa�w�b�S�efore�using�norms,���the�ring��O�����K�� ��is� nitely�generated�as�a�mo�dule�����o��rv�er����src:40 day4.tex�Z�,��$so�it�is�certainly�nitely�generated�as�a�ring�o��rv�er����src:41 day4.tex�Z�.�N�By�the�Hilb�S�ert�Basis�����Theorem,���O�����K��&j�is�No�S�etherian.���m��cff���x�ff̟���ff̎�̄�cff�������%��src:44 day4.texIf�?p���g�cmmi12�R�X��is�an�in��rtegral�domain,�T�the��eld��"of�fr��ffactions�?p�of��R��is�the�eld�of�all�elemen��rts������src:45 day4.tex�a=b�,��'where���a;���b�UR�2��R�J�.�.�The�eld�of�fractions�of��src:46 day4.tex�R��P�is�the�smallest�eld�that�con��rtains��R��.�����F��Vor��example,�the�eld�of�fractions�of��src:47 day4.tex�Z��is��Q��and�of��src:48 day4.tex�Z�[(1���+�������p�������ǉz����	�9��5������)�=�2]��is��Q�(������p���
��ǉz����	�9��5������).������Denition���1.2�(In��tegrally�Closed).���src:51 day4.tex�An���in��rtegral�domain��R�(�is��inte��ffgr�al���ly�6)close�d�in�����its��eld�of�fr��ffactions�m��if�whenev��rer��src:52 day4.tex����is�in�the�eld�of�fractions�of��R��йand��src:53 day4.tex���satises�a�����monic��p�S�olynomial��f��Q�2�UR�R�J�[�x�],�then���h��2��R�J�.������Prop�osition��01.3.���src:58 day4.tex�If�?\�K���is�any�numb��ffer�eld,�Bethen��O�����K��{�is�inte�gr�al���ly�close�d.���A��2lso,�Bethe�����ring�35�src:59 day4.tex���S��z�@�	�\��Z���s5�of�al���l�algebr��ffaic�inte�gers�is�inte�gr�al���ly�close�d.�������1����*��a����L���C�����퍍����Pr��ffo�of.���8I�src:62 day4.tex�W��Ve�$# rst�pro��rv�e�$#that����S��z�@�	�\��Z����F�is�in��rtegrally�closed.��PSupp�S�ose��c��(�2����S��z�
#��	�\��Q����عis�$#in�tegral�o�v�er��src:63 day4.tex���S��z�@� �\��Z���@�,������so�E;there�is�a�monic�p�S�olynomial��f�G��(�x�)��|=��x����2�n�����+��S�a�����n�#�K� cmsy8�� |{Ycmr8�1�����x����2�n��1��m �+���������Ф�+��a�����1����x��+��a�����0�� ?�with�E;�src:64 day4.tex�a�����i��TV�2���|��S��z�@� �\��Z��������and���src:65 day4.tex�f�G��(�c�)� m=�0.�>�The��a�����i��Q|�all�lie�in�the�ring�of�in��rtegers��O�����K�� (d�of�the�n�um�b�S�er� eld��src:66 day4.tex�K�� �=������Q�(�a�����0����;���a�����1���;��:�:�:��ʜa�����n��1���̹),��Land����O�����K�� ,��is� nitely�generated�as�a��src:67 day4.tex�Z�-mo�S�dule,�so��Z�[�a�����0����;����:�:�:��ʜ;���a�����n��1���̹]�is����� nitely��mgenerated�as�a��src:68 day4.tex�Z�-mo�S�dule.��wSince��f�G��(�c�)�UR=�0,��w��re��mcan�write��c����2�n��t��as�a��src:69 day4.tex�Z�[�a�����0����;����:�:�:��ʜ;���a�����n��1���̹]-�����linear�_&com��rbination�of��c����2�i����for��i��<�n�,�|Eso�_&the�ring��src:70 day4.tex�Z�[�a�����0����;����:�:�:��ʜ;���a�����n��1�����;�c�]�_&is�also� nitely�gen-�����erated���as�a��src:71 day4.tex�Z�-mo�S�dule.�[RTh��rus��Z�[�c�]�is� nitely�generated�as��Z�-mo�dule�b�ecause�it�is�a�����submo�S�dule��of�a� nitely�generated��src:72 day4.tex�Z�-mo�dule,���whic��rh�implies�that��src:73 day4.tex�c��is�in�tegral�o�v�er��Z�.����% ��src:75 day4.texSupp�S�ose�؄�c��0�2��K��"�is�in��rtegral�o�v�er��O�����K��;¹.�tThen�since����S��z�@� �\��Z�����is�in�tegrally�closed,��src:76 day4.tex�c��is�an�����elemen��rt��of����S��z�@� �\��Z��� *��,�so��c�UR�2��K��F�\������S��z�@� �\��Z���?��=��O�����K��;¹,��as�required.���� ���cff���x�ff ̟���ff ̎� ̄�cff���������De nition���1.4�(Dedekind�Domain).�fg�src:81 day4.tex�An�" in��rtegral�domain��R�;j�is�a��De��ffdekind�z�domain������if���it�is�No�S�etherian,���in��rtegrally�closed�in�its� eld�of�fractions,�and�ev��rery�nonzero�prime�����ideal��of��src:83 day4.tex�R��is�maximal.����% ��src:85 day4.texThe�~ring��Q�ü���Q�~�is�No�S�etherian,��in��rtegrally�closed�in�its� eld�of�fractions,�and�the�����t��rw�o�R prime�ideals�are�maximal.�oHHo��rw�ev�er,�k�it�R is�not�a�Dedekind�domain�b�S�ecause�it�is�����not��xan�in��rtegral�domain.�3�The�ring��src:88 day4.tex�Z�[������p��� ��ǉz���� �9��5������]�is�not�a�Dedekind�domain�b�S�ecause�it�is�not�����in��rtegrally�J�closed�in�its� eld�of�fractions,�b�as��src:89 day4.tex(1��+�������p��� ���ǉz���� �9��5����� )�=�2�J�is�in�tegrally�o�v�er��src:90 day4.tex�Z��and�lies�����in��e�Q�(������p��� ��ǉz���� �9��5������),��Tbut�not�in��Z�[������p��� ��ǉz���� �9��5����].��The�ring��src:91 day4.tex�Z��is�a�Dedekind�domain,��Tas�is�an��ry�ring�of�����in��rtegers����O�����K�� �Q�of�a�n�um�b�S�er� eld,���as�w�e�will�see�b�S�elo�w.�$-Also,���an�y�eld��src:92 day4.tex�K��-�is�a�Dedekind�����domain,��}since�lit�is�a�domain,�it�is�trivially�in��rtegrally�closed�in�itself,�and�there�are�����no��nonzero�prime�ideals�so�that�condition�that�they�b�S�e�maximal�is�empt��ry��V.������Prop�osition��1.5.����src:98 day4.tex�The��<ring�of�inte��ffgers��O�����K��!��of�a�numb�er�eld�is�a�De�dekind�domain.�������Pr��ffo�of.���8I�src:101 day4.tex�By�&EProp�S�osition�1.3,�5-the�ring��O�����K��b�is�in��rtegrally�closed,�and�b��ry�Prop�S�osition�1.1�����it���is�No�S�etherian.�pSupp�ose���that��src:103 day4.tex�3�%n�
eufm10�p��is�a�nonzero�prime�ideal�of��O�����K��;¹.�Let��src:104 day4.tex�����2��"�p��b�S�e�a�����nonzero��elemen��rt,�and�let��f�G��(�x�)�UR�2��Z�[�x�]��b�S�e�the�minimal�p�olynomial�of��src:105 day4.tex����.�8�Then�����y @�f�G��(����)�UR=��������n��f��+����a�����n��1�����������n��1��C�+���������UN�+��a�����1������7�+��a�����0��V�=�UR0�;������so���src:107 day4.tex�a�����0��	v�=�Zr��(�������2�n�����+��a�����n��1�����������2�n��1���v�+���������&4�+��a�����1�������)��2��p�.�Since���src:108 day4.tex�f���is�irreducible,��l�a�����0��	D�is�a�nonzero������elemen��rt��
of��Z��that�lies�in��src:109 day4.tex�p�.�}Ev�ery�elemen�t�of�the�nitely�generated�ab�S�elian�group������src:110 day4.tex�O�����K��;��=�p�.��is�killed�b��ry��a�����0����,�?�so��O�����K���=�p��is�a�nite�set.��Since��src:111 day4.tex�p��is�prime,�?��O�����K���=�p��is�an�in��rtegral�����domain.���Ev��rery��Tnite�in�tegral�domain�is�a�eld,��so��src:112 day4.tex�p��is�maximal,�whic��rh�completes�����the��pro�S�of.���r>���cff���x�ff̟���ff̎�̄�cff�������%��src:116 day4.texIf��w�I����and��J���are�ideals�in�a�ring��R�J�,��+the�pro�S�duct��I��J��is�the�ideal�generated�b��ry�all�����pro�S�ducts��of�elemen��rts�in��src:117 day4.tex�I��+�with�elemen�ts�in��J�r�:������7]�I��J�qĹ=�UR(�ab��:��a��2��I�;���b��2��J�r�)����R�J:������Note�that�the�set�of�all�pro�S�ducts��src:121 day4.tex�ab�,�
hwith��a����2��I����and��b��2��J�r�,�need�not�b�S�e�an�ideal,�so������it���is�imp�S�ortan��rt�to�tak�e�the�ideal�generated�b�y�that�set.�9(See�the�homew�ork�problems�����for��examples.)������2������a����L���C�����퍑��Denition�\W1.6�(F���ractional�Ideal).��:�src:126 day4.tex�A�˗�fr��ffactional��ide�al�˞�is�an��O�����K��;¹-submo�S�dule�of��I�F���������K��F�that��is�nitely�generated�as�an��src:127 day4.tex�O�����K��;¹-mo�S�dule.�����%��src:129 day4.texT��Vo��Va��rv�oid�confusion,��Bw�e�will�sometimes�call�a�gen�uine�ideal��I��I����O�����K����an��inte��ffgr�al�����ide��ffal�.���Also,��since��fractional�ideals�are�nitely�generated,�w��re�can�clear�denominators�����of��a�generating�set�to�see�that�ev��rery�fractional�ideal�is�of�the�form��src:132 day4.tex�aI�Fչ=�UR�f�ab��:��b��2��I���g������for��some��src:133 day4.tex�a�UR�2��K��F�and��ideal��I�F���URO�����K��;¹.����%��src:135 day4.texF��Vor�ˇexample,���the�collection�����Fu�����1��������z�@���2�����
q��Z��of�rational�n��rum�b�S�ers�ˇwith�denominator��src:136 day4.tex1�or�2�is�a�����fractional��ideal�of��Z�.������Theorem���1.7.�7\�src:139 day4.tex�The���set�of�nonzer��ffo�fr�actional�ide�als�of�a�De�dekind�domain��R��%�is�an�����ab��ffelian�35gr�oup�under�ide�al�multiplic�ation.����%��src:142 day4.tex�Before�:�pro��rving�Theorem�1.7�w�e�pro�v�e�a�lemma.�(�F��Vor�the�rest�of�this�section��src:143 day4.tex�O�����K�������is��the�ring�of�in��rtegers�of�a�n�um�b�S�er�eld��src:144 day4.tex�K�ܞ�.������Denition�T|1.8�(Divides�for�Ideals).����src:148 day4.tex�Supp�S�ose���that��I��;���J��:�are�ideals�of��O�����K��;¹.�,@Then��src:149 day4.tex�I������divides���J��if��I�F���UR�J�r�.����%��src:151 day4.texT��Vo���see�that�this�notion�of�divides�is�sensible,��	supp�S�ose��K��s�=���Q�,�so��src:152 day4.tex�O�����K��Q��=���Z�.�����Then�E�I�ࢹ=��(�n�)�and��J���=�(�m�)�for�some�in��rteger��n��and��m�,�[�and��src:153 day4.tex�I�6��divides��J�av�means�that�����(�n�)�UR���(�m�),�o�i.e.,�that�Qthere�exists�an�in��rteger��src:154 day4.tex�c��suc�h�that��m�UR�=��cn�,�o�whic�h�Qexactly�means�����that���n��divides��src:155 day4.tex�m�,�as�exp�S�ected.������Lemma��t1.9.���src:158 day4.tex�Supp��ffose�;��I�-�is�an�ide�al�of��O�����K��;��.��Then�ther�e�exist�prime�ide�als��src:159 day4.tex�p�����1����;����:�:�:��ʜ;����p�����n�������such�5�that��p�����1��l������p�����2��������������p�����n��	l��Z�I���.�n+In�other�wor��ffds,��src:160 day4.tex�I�'N�divides�a�pr�o�duct�of�prime�ide�als.�n+(By�����c��ffonvention��Lthe�empty�pr�o�duct�is�the�unit�ide�al.�AA��2lso,�ٮif��src:161 day4.tex�I�Fչ=�UR0�,�then�we�take��src:162 day4.tex�p�����1��V�=�UR(0)�,�����which�35is�a�prime�ide��ffal.)�������Pr��ffo�of.���8I�src:165 day4.tex�The���k��rey�idea�is�to�use�that��O�����K��
6��is�No�S�etherian�to�deduce�that�the�set��src:166 day4.tex�S����of�����ideals�R�that�do�not�satisfy�the�lemma�is�empt��ry��V.�pkIf��S�X�is�nonempt�y��V,�lwthen�b�S�ecause��src:167 day4.tex�O�����K�������is���No�S�etherian,��fthere�is�an�ideal��I�J��2�Y�S����that�is�maximal�as�an�elemen��rt�of��src:168 day4.tex�S��׹.�?uIf��I��]�w�ere�����prime,�+�then��g�I���w��rould�trivially�con�tain�a�pro�S�duct�of�primes,�+�so��src:169 day4.tex�I���is�not�prime.�;By�����denition���of�prime�ideal,�E0there�exists��src:170 day4.tex�a;���b�-0�2�O�����K��
;��suc��rh���that��ab�-0�2��I��d�but����src:171 day4.tex�a��62��I��and������b����62��I��.��@Let�s�J�����1��S��=��I��8�+�õ(�a�)�and��src:172 day4.tex�J�����2��S��=��I��8�+�õ(�b�).��@Then�neither��J�����1���w�nor��J�����2���is�in��S��׹,��since��I���is�����maximal,��xso���b�S�oth��src:173 day4.tex�J�����1��H�and��J�����2���con��rtain�a�pro�S�duct�of�prime�ideals.�LTh�us�so�do�S�es��src:174 day4.tex�I��,��xsince��������J�����1����J�����2��V�=�UR�I������2��\/�+����I��(�b�)�+�(�a�)�I��+�+�(�ab�)����I��;������whic��rh��is�a�con�tradiction.�8�Th�us��src:176 day4.tex�S���is�empt�y��V,�whic�h�completes�the�pro�S�of.���;˄�cff���x�ff̟���ff̎�̄�cff�������%��src:179 day4.texW��Ve��are�no��rw�ready�to�pro�v�e�the�theorem.��������Pr��ffo�of�35of�The��ffor�em�351.7.�������src:182 day4.tex�The��ypro�S�duct�of�t��rw�o��yfractional�ideals�is�again�nitely�gener-������ated,��so��it�is�a�fractional�ideal,�and��src:183 day4.tex�I���O�����K���[�=�~��O�����K��>��for�an��ry�nonzero�ideal��I��,��so�to�pro��rv�e�����that���the�set�of�fractional�ideals�under�m��rultiplication�is�a�group�it�suces�to�sho�w�����the���existence�of�in��rv�erses.�:#W��Ve���will�rst�pro��rv�e���that�if��src:186 day4.tex�p��is�a�prime�ideal,���then��p��has������3����$��a����L���C�����퍑��an��fin��rv�erse,��@then�w�e�will�pro�v�e�that�nonzero�in�tegral�ideals�ha�v�e�in�v�erses,��@and� nally������observ��re��that�ev�ery�fractional�ideal�has�an�in�v�erse.����% ��src:190 day4.texSupp�S�ose���p��is�a�nonzero�prime�ideal�of��O�����K��;¹.�8�W��Ve�will�sho��rw�that�the��src:191 day4.tex�O�����K���-mo�S�dule��� ���>&�I�Fչ=�UR�f�a��2��K�1�:��a�p���O�����K��;��g������is��a�fractional�ideal�of��src:195 day4.tex�O�����K�� &j�suc��rh�that��I���p�UR�=��O�����K��;¹,��so�that��src:196 day4.tex�I��+�is�an�in�v�erse�of��p�.�����% ��src:198 day4.texF��Vor��$the�rest�of�the�pro�S�of,���x�a�nonzero�elemen��rt��b�\�2��p�.�LUSince��$�I�⧹is�an��src:199 day4.tex�O�����K��;¹-mo�dule,������bI������@O�����K�� ��is�jAan��O�����K���ideal,��(hence��I�[Ĺis�a�fractional�ideal.� ��Since��src:200 day4.tex�O�����K�� ���@�I��w��re�ha�v�e������p������I���p���O�����K��;¹,�vhence�ͳeither��src:201 day4.tex�p��ǹ=��I��p�ͳ�or��I��p��ǹ=��O�����K��;¹.��If�ͳ�src:202 day4.tex�I��p��=��O�����K��;¹,�vw��re�ͳare�done�since�����then�MW�I�>ڹis�an�in��rv�erse�MWof��p�.��Th��rus�supp�S�ose�that��src:203 day4.tex�I���p��K�=��p�.�Our�MWstrategy�is�to�sho��rw�that�����there���is�some��src:204 day4.tex�d�nr�2��I��U�not���in��O�����K��;¹;��fsuc��rh�a��d��w�ould�lea�v�e��p��in�v��X�arian�t�(i.e.,�� �src:205 day4.tex�d�p�nr���p�),�so�����since���p��is�an��O�����K��;¹-mo�S�dule�it�will�follo��rw�that��src:206 day4.tex�d�UR�2�O�����K���,��a�con��rtradiction.����% ��src:208 day4.texBy��Lemma�1.9,�w��re�can�c�ho�S�ose�a�pro�duct��p�����1����;����:�:�:��ʜ;����p�����m��Ĺ,�with��src:209 day4.tex�m��minimal,�suc��rh�that��� ������p�����1����p�����2��������������p�����m�� Z��UR�(�b�)����p�:������If�Gno��src:212 day4.tex�p�����i���!�is�con��rtained�in��p�,�'othen�w�e�can�c�ho�S�ose�for�eac�h��i��an��src:213 day4.tex�a�����i�� ��2���p�����i���!�with��a�����i���62���p�;�3�but������then���������%��u cmex10�Q���R�a�����i���v�2�X��p�,��whic��rh���con�tradicts�that��src:214 day4.tex�p��is�a�prime�ideal.��Th�us�some��p�����i��dڹ,��sa�y��src:215 day4.tex�p�����1����,��is�����con��rtained�Rin��p�,�A�whic�h�implies�that��p�����1��V�=�UR�p��since�ev�ery�nonzero�prime�ideal�is�maximal.�����Because�ve�src:216 day4.tex�m��is�minimal,��T�src:217 day4.tex�p�����2��������������p�����m�� {)�is�not�a�subset�of�(�b�),�so�there�exists��c�C,�2��p�����2��������������p�����m�������that� Mdo�S�es�not�lie�in��src:218 day4.tex(�b�).���Then��p�(�c�)��{���(�b�),��so� Mb��ry�de nition�of��src:219 day4.tex�I��йw�e�ha�v�e��d��{�=��c=b��2��I��.�����Ho��rw�ev�er,�k��d�7�62�O�����K��;¹,�since�Q�if�it�w��rere�then��src:220 day4.tex�c��w�ould�b�S�e�in�(�b�).�n�W��Ve�ha�v�e�th�us�found�our�����elemen��rt��1�src:221 day4.tex�d��:�2��I����that�do�S�es�not�lie�in��O�����K��;¹.�nzT��Vo� nish�the�pro�of�that��src:222 day4.tex�p��has�an�in��rv�erse,�����w��re��observ�e�that��d��preserv�es�the��src:223 day4.tex�O�����K��;¹-mo�S�dule��p�,���and�is�hence�in��O�����K���,���a�con��rtradiction.�����More�AHprecisely��V,�c(if��src:224 day4.tex�b�����1����;����:�:�:��ʜ;���b�����n��阹is�a�basis�for��p��as�a��Z�-mo�S�dule,�then�the�action�of��src:225 day4.tex�d��on��p��is�����giv��ren���b�y�a�matrix�with�en�tries�in��src:226 day4.tex�Z�,��mso�the�minimal�p�S�olynomial�of��d��has�co�ecien��rts�����in����Z�.� �This�implies�that��src:227 day4.tex�d��is�in��rtegral�o�v�er��Z�,���so��d�UR�2�O�����K��;¹,�since����O�����K�� �I�is�in��rtegrally�closed�����b��ry�ڔProp�S�osition�1.3.�3�(Note�ho�w�this�argumen�t�dep�S�ends�strongly�on�the�fact�that��src:229 day4.tex�O�����K�������is��in��rtegrally�closed!)����% ��src:232 day4.texSo���far�w��re�ha�v�e�pro�v�ed�that�if��p��is�a�prime�ideal�of��O�����K��;¹,�"!then��src:233 day4.tex�p����2��1��Y�=��t�f�a��2��K��:������a�p�����O�����K��;��g�n �is�the�in��rv�erse�n of��p��in�the�monoid�of�nonzero�fractional�ideals�of��src:234 day4.tex�O�����K���.�����As��Smen��rtioned�after�De nition�1.6,��=ev�ery�nonzero�fractional�ideal�is�of�the�form��src:236 day4.tex�aI������for����a�X��2��K��{�and��I���an�in��rtegral�ideal,�!*so�since�(�a�)�has�in�v�erse��src:237 day4.tex(1�=a�),�!*it�suces�to�����sho��rw���that�ev�ery�in�tegral�ideal��I�w'�has�an�in�v�erse.� �If�not,��cthen�there�is�a�nonzero�����in��rtegral�-�ideal��src:238 day4.tex�I� :�that�is�maximal�among�all�nonzero�in�tegral�ideals�that�do�not�ha�v�e�����an�� in��rv�erse.�IEv�ery�ideal�is�con�tained�in�a�maximal�ideal,��aso�there�is�a�nonzero�prime�����ideal�#-�src:241 day4.tex�p��suc��rh�that��I�Z���iD�p�.��nThen��I���iD�p����2��1�� \|�I���O�����K��;¹.��nIf�#-�src:242 day4.tex�I��=��p����2��1�� \|�I��,�qNthen�#-(arguing�as�����in�3Ithe�previous�paragraph)�eac��rh�elemen�t�of��src:243 day4.tex�p����2��1���Źpreserv�es�that��src:244 day4.tex�O�����K��;¹-ideal��I�$̹and�is�����hence���in��rtegral,���so��p����2��1������XO�����K��;¹,�whic��rh�implies�that��src:245 day4.tex�O�����K���ܹ=�X�pp����2��1�������p�,�a���con��rtradiction.�����Th��rus��q�src:246 day4.tex�I����6�=���p����2��1��\|�I��.��<Because��I���is�maximal�among�ideals�that�do�not�ha�v�e�an�in�v�erse,�����the���ideal��src:247 day4.tex�p����2��1��\|�I��3�do�S�es�ha��rv�e���an�in��rv�erse,�D�call���it��src:248 day4.tex�J�r�.�w�Then��p�J�"�is�the�in��rv�erse���of��I��,�since������O�����K����=�UR(�p�J�r�)(�p����2��1��\|�I��)�=��J�I��.���*����cff���x�ff̟���ff̎�̄�cff��������%��src:252 day4.texW��Ve�e�can�nally�deduce�the�crucial�Theorem�1.11,��-whic��rh�will�allo�w�us�to�sho�w�that�����an��ry��nonzero�ideal�of�a�Dedekind�domain�can�b�S�e�expressed�uniquely�as�a�pro�duct������4����6��a����L���C�����퍑��of���primes�(up�to�order).���Th��rus�unique�factorization�holds�for�ideals�in�a�Dedekind������domain,��Cand���it�is�this�unique�factorization�that�initially�motiv��X�ated�the�in��rtro�S�duction�����of��rings�of�in��rtegers�of�n�um�b�S�er�elds�o�v�er�a�cen�tury�ago.���3����Theorem��1.10.�2<�src:260 day4.tex�Supp��ffose����I�%�is�an�inte�gr�al�ide�al�of��O�����K��;��.�u�Then��I�%�c�an�b�e�written�as�a�����pr��ffo�duct������E�I�Fչ=�UR�p�����1��������������p�����n��������of�*�prime�ide��ffals�of��src:264 day4.tex�O�����K��;��,�,and�this�r�epr�esentation�is�unique�up�to�or�der.�c�(Exc�eption:�b$If������src:265 day4.tex�I�Fչ=�UR0�,�35then�the�r��ffepr�esentation�35is�not�unique.)�������Pr��ffo�of.���8I�src:268 day4.tex�Supp�S�ose��C�I�zƹis�an�ideal�that�is�maximal�among�the�set�of�all�ideals�in��src:269 day4.tex�O�����K�������that��can�not�b�S�e�written�as�a�pro�duct�of�primes.� oEv��rery�ideal�is�con�tained�in�a�����maximal�k�ideal,�˷so��src:270 day4.tex�I�]�is�con��rtained�in�a�nonzero�prime�ideal��src:271 day4.tex�p�.� �jIf��I���p����2��1��@ܹ=���I��,�then�����b��ry��Theorem�1.7�w�e�can�cancel��src:272 day4.tex�I� R�from�b�S�oth�sides�of�this�equation�to�see�that��src:273 day4.tex�p����2��1��x�=������O�����K��;¹,�z�a�*�con��rtradiction.���Th�us��I� *�is�strictly�con�tained�in��src:274 day4.tex�I���p����2��1�� \|�,�z�so�b�y�our�maximalit�y�����assumption��ton��I����there�are�maximal�ideals��src:275 day4.tex�p�����1����;����:�:�:��ʜ;����p�����n�� BĹsuc��rh�that��I���p����2��1���ƹ=�4J�p�����1������������p�����n���P�.�����Then���src:276 day4.tex�I�Fչ=�UR�p�/����p�����1��������������p�����n���P�,��na�con��rtradiction.�$�Th�us�ev�ery�ideal�can�b�S�e�written�as�a�pro�duct�����of��primes.����%��src:279 day4.texSupp�S�ose���p�����1��������������p�����n��	Cֹ=����q�����1������������q�����m��Ĺ.���If�no��q�����i��x��is�con��rtained�in��src:280 day4.tex�p�����1����,�6then�for�eac�h��i��there�is�����an�SG�a�����i��[email protected]�2�f�q�����i���!�suc��rh�that��src:281 day4.tex�a�����i���62�f�p�����1����.�r�But�the�pro�S�duct�of�the��a�����i���!�is�in�the��p�����1��������������p�����n���P�,�mowhic��rh�is�����a�9subset�of��src:282 day4.tex�p�����1����,��whic��rh�con�tradicts�the�fact�that��src:283 day4.tex�p�����1���=�is�a�prime�ideal.���Th�us��q�����i����=��D�p�����1���=�for�����some����i�.�
OW��Ve�can�th��rus�cancel��src:284 day4.tex�q�����i��맹and��p�����1��	Fѹfrom�b�S�oth�sides�of�the�equation.�Rep�S�eating�����this��argumen��rt�nishes�the�pro�S�of�of�uniqueness.�������cff���x�ff̟���ff̎�̄�cff��������Corollary��d1.11.���src:289 day4.tex�If���I��=�is�a�fr��ffactional�ide�al�of��O�����K��B|�then�ther�e�exists�prime�ide�als������src:290 day4.tex�p�����1����;����:�:�:��ʜ;����p�����n��	ۅ�and�35�q�����1���;��:�:�:��ʜ;��q�����m����,�35unique�up�to�or��ffder,�such�that��������I�Fչ=�UR(�p�����1��������������p�����n���P�)(�q�����1������������q�����m��Ĺ)������1��\|�:��������Pr��ffo�of.���8I�src:297 day4.tex�W��Ve�lha��rv�e��I��ӹ=��P(�a=b�)�J����for�some��a;���b��2�O�����K��
�йand�in��rtegral�ideal��J�r�.�	�Applying������Theorem�ce1.11�to��src:298 day4.tex(�a�),���(�b�),�and�ce�J�׹giv��res�an�expression�as�claimed.��F��Vor�uniqueness,���if�����one��!has�t��rw�o��!suc�h�pro�S�duct�expressions,��m�ultiply�through�b�y�the�denominators�and�����use��the�uniqueness�part�of�Theorem�1.11����l��cff���x�ff̟���ff̎�̄�cff����'I����src:304 day4.tex��2��(�Using�z�MA��u�GMA��b#����src:305 day4.tex�This��;section�is�a�rst�in��rtro�S�duction�to�MA�GMA,�whic�h�is�an�excellen�t�pac�k��X�age�for�����doing��Falgebraic�n��rum�b�S�er��Ftheory�computations.�
��Y��Vou�can�use�it�via�the�w��reb�page������6߆�Tcmtt12�http://modular.fas.harvard.edu/calc�.��MA��rGMA��Qis�͌not�free,�Ebut�if�y�ou�w�ould�����lik��re���a�cop�y�for�y�our�p�S�ersonal�computer,���send�me�an�email,�and�I���can�arrange�for�y��rou�����to��obtain�a�legal�cop��ry�for�free.��1(Sa�y�something�ab�S�out�m�y�visiting�MA�GMA�in�Sydney�����three��times,�and�ho��rw�MA�GMA�compares�to�Maple,�Mathematica,�and�P��VARI.)��R���!�1.����0ߞ�src:314 day4.texMA��rGMA��w�eb�page������!�2.����0ߞ�src:318 day4.texExample���co�S�de�to�illustrate�things�so�far�in�course,��and�relev��X�an��rt�to�eac�h�home-����0ߞw��rork��problems.�8�Exp�S�erimen�t�with�studen�ts�suggesting�what�examples�to�try��V.������5����N���a����L���C�����퍑��src:324 day4.tex��3��(�Algorithms�z�for�Algebraic�Num��u�b��=er�Theory��b#����src:325 day4.tex�The�7b�S�est�o��rv�erall�7reference�for�algorithms�for�doing�basic�algebraic�n��rum�b�er�7theory������computations���is�Henri�Cohen's�b�S�o�ok����A�<�Course�<�in�Computational�A��2lgebr��ffaic�Numb�er�����The��ffory�,��Springer,�GTM�138.����%��src:329 day4.texOur�[�main�long-term�algorithmic�goals�for�this�course�are�to�understand�go�S�o�d�����algorithms��for�solving�the�follo��rwing�problems�in�particular�cases:��������$�������0ߞ�src:333 day4.tex�Ring���of�in��tegers:���Giv��ren�9�a�n�um�b�S�er� eld��K���(b�y�giving�a�p�S�olynomial),�]?compute����0ߞthe��full�ring��src:334 day4.tex�O�����K�� &j�of�in��rtegers.�������$�������0ߞ�src:335 day4.tex�Decomp�osition��of�primes:�?��Giv��ren�na�prime�n�um�b�S�er��p�5�2��Z�,���nd�nthe�decom-����0ߞp�S�osition��of�the�ideal��src:336 day4.tex�p�O�����K��&j�as�a�pro�duct�of�prime�ideals�of��src:337 day4.tex�O�����K��;¹.�������$�������0ߞ�src:338 day4.tex�Class���group:�f�Compute�Q�the�group�of�equiv��X�alence�classes�of�nonzero�ideals�of����0ߞ�src:339 day4.tex�O�����K��;¹,�Y�where�5k�I�&�and��J�Qݹare�equiv��X�alen��rt�if�there�exists��src:340 day4.tex� �h��2�URO�����K�� q-�suc�h�that��src:341 day4.tex�I��J�� r���2��1���@�=�UR(� ���).�������$�������0ߞ�src:342 day4.tex�Units:�8�Compute��generators�for�the�group�of�units�of��src:343 day4.tex�O�����K��;¹.����%��src:346 day4.texAs�
�w��re�will�see,�R�somewhat�surprisingly�it�turns�out�that�algorithmically�b�y�far�����the��+most�time-consuming�step�in�computing�the�ring�of�in��rtegers��src:348 day4.tex�O�����K���is�to�factor�the�����discriminan��rt���of�a�p�S�olynomial�whose�ro�ot�generates�the�eld��src:349 day4.tex�K�ܞ�.�LThe�algorithm(s)�for�����computing�
L�src:350 day4.tex�O�����K��I�are�quite�complicated�to�describ�S�e,��but�the�rst�step�is�to�factor�this�����discriminan��rt,�74and��it�tak�es�m�uc�h�longer�in�practice�than�all�the�other�complicated�����steps.������6����a����;��a���N�6߆�Tcmtt12�3�%n�
eufm10�0��N�cmbx12�/��N�G�cmbx12�%��u
cmex10�#�K�cmsy8�"!",�
cmsy10� �2cmmi8���g�cmmi12�|{Ycmr8����@cmti12�X�Qffcmr12���N��qcmbx12�D��t�qG�cmr17�X�Qcmr12�j��������`